# International Mathematics Competition for University Students 2016

Select Year:

IMC 2019
 Information Results/Prizes Problems & Solutions Photos

## IMC2016: Day 1, Problem 4

4. Let $n\ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties:

(i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets containing exactly $k$ elements;

(ii) for any two sets $A,B\in \mathcal{F}$, their union $A\cup B$ also belongs to $\mathcal{F}$.

Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements.

Proposed by Fedor Petrov, St. Petersburg State University

 IMC1994 IMC1995 IMC1996 IMC1997 IMC1998 IMC1999 IMC2000 IMC2001 IMC2002 IMC2003 IMC2004 IMC2005 IMC2006 IMC2007 IMC2008 IMC2009 IMC2010 IMC2011 IMC2012 IMC2013 IMC2014 IMC2015 IMC 2016 IMC2017 IMC2018 IMC2019